1. Use the field axioms and their consequences to prove the following properties of real numbers. Your proofs should be careful and complete, with only one axiom, theorem, or other justification per step.
(a) For any two real numbers a and b, we have that if a + b = 0, then b = -a. (Uniqueness of the additive inverse.)
(b) For all a ∈ R, we have (-1)a = -a.
(c) (-1). (-1) = 1.
(d) For all a, b∈ R, we have (-a). (-b) = a b.
2. Use the field axioms, order axioms, and their consequences to prove the following properties of real numbers. (Algebraic properties will still need to be carefully proved/justified as you use them.)
(a) For all a, b, c, d ∈ R, if a < b andc≤ d, then a + c< b + d.
(b) If a, b ∈ R with 0 < a < b, then lib < 1/a.
3. Warm-up problems. These problems are just meant to be quick warm-ups to get you thinking about the definitions introduced in class and the reading. You do not need to justify anything or prove your answers.
(a) -(c) Find the infima and suprema of the set A = {1 + (-2/3)n ¦n ∈ N}, the function B given by B(x) = 4x - T2, and the set C = {1/n - 1/m ¦n, m ∈ N}.
(4) Give an example of a finite set D ⊆ R with sup D = -7 and inf D = -20.
4. State and prove a counterpart to Lemma 2.3.4 for inf S.
5. Suppose T ⊆ S ⊆ R are nonempty and bounded. Prove inf S ≤ muff' ≤ sup T ≤ sup S. Hint: Prove each inequality separately.